Note that the solution to the transport equation is constant on any straight line of the form in the plane. These straight lines are called the base characteristic curves. The equation defines a plane in three dimensions. The intersections of these planes with the solution surface are called characteristic curves. Since the characteristic curves are solutions to a system of ODEs, the problem of solving the PDE is reduced to that of solving a system of ODEs for , , and , where is a parameter along the characteristic curves. These ODEs are called characteristic ODEs.

The solution to an inhomogeneous PDE has two components: the general solution to the homogeneous PDE and a particular solution to the inhomogeneous PDE.

A practical consequence of quasi-linearity is the appearance of shocks and steepening and breaking of solutions. Thus, although the procedures for finding general solutions to linear and quasi-linear PDEs are quite similar, there are sharp differences in the nature of the solutions.